Axisymmetric laminar blood flows in a stenotic model vessel are calculated by using a finite-difference scheme of the Navier-Stokes equations. The wavy lumen are transformed to a rectangular region with fine and irregular meshes mapped to regular ones in the computational domain. The pulsating flow is controlled by a time-dependent von Kármán number. At the upstream and downstream ends, the local acceleration is employed to replace the Neumann boundary condition for solving the pressure field from the divergence of the Navier-Stokes equations, resulting in a new treatment on the inlet and outlet conditions for transient flow problems. The kinematic and dynamic characteristics of vortices in the pulsating flow processes are analyzed along with the pressure and shear fields.